Equation of State
Michael Palmer
Equations of State
• Common ones we’ve heard of
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Ideal gas
Barotropic
Adiabatic
Fully degenerate
• Focus on fully degenerate EOS
• Specifically on corrections made to this EOS
White Dwarfs
• Physics of WD’s provide a lot of
opportunity to improve EOS of a fully
degenerate gas
• Fully degenerate core
• Partially degenerate towards surface
• Crystallization (starting in the interior)
• Physics of crystallization
Crystallization
• Interior of WD, ions can not be treated as
ideal
• Must consider Coulomb interactions
• Electrons not effect at screening ions 
favourable for ions to rearrange themselves
• When 3kT/2 is of the order of -Ze
• 3kT/2 => Thermal energy
• -Ze => Coulomb energy per ion
Crystallization
• Coulomb coupling ratio
•  = (Ze)2 / (rikT) (Kippenhahn and Weigert, pg 134)
– ri is the mean separation between ions
» Defined as (3/(4ni))1/3
» ni is the number density of ions
– k is the Boltzmann constant and T is temperature
• Gives insight into the strength of the Coulomb
interactions
• If  >> 1, kinetic energy’s role not significant and ions will
try and settle into a lower energy state
• Crystallization at  ≈ 175 for one component
plasma
• Intermediate values results in phase transition from a gas
to a liquid
Crystallization
• Tm ≈ (Z2e2/ ck)(4/30mu)1/3 critical
temperature obtained from  and density
relation (Kippenhahn and Weigert, pg 134)
• Phase from liquid to solid can not be gradual
• Symmetry properties
• First order phase transition => lose latent heat
• Phase transition found to be first order
(Winget et al. 2009) PUT IN FIGURE!!!!
• Latent heat ~ kT per ion, slows cooling
• Observed as bump => PUT IN FIGURE!!!!
Crystallization
• Lattice ions oscillate
• Coulomb energy -EC = 2Z/(A1/3)61/3keV
» As T-> 0 ions not at rest
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Oscillate about points of equilibrium
Frequency E2 ~ Z2e2n0/m0
ZEzp = 3Eh(bar)/2 and Ezp = (0.6/A)61/2keV
Ezp << EC so does not contribute as much to E = E0 + EC
+ Ezp ~ E0 + EC
• Find EC influences the pressure by lowering it as
compared to an ideal Fermi gas
– From P  -dE / d(1/n)
Crystallization
• Specific heat Cv effect
• When  << 1 the ions in the interior of the white
dwarf behave as an ideal gas, Cv = 3k/2
• As ions form lattice, energy goes into lattice
oscillations, results in additional degrees of
freedom which raise Cv to a maximum of 3k
•  = 2cvMT / 5L
Crystallization
• Electron polarization, Coulomb crystal
• fie = -f∞(xr)[1 + A(xr)(Q()/ )8]
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fie is the correction to free energy
f∞(xr) = b1√(1 + b2/x2)
A(xr) = (b3 + a3x2) / (1 + b4x2)
Q() classically defined as ≈ q
 is TP/T
xr = pF/mec
b1,b2,b3,b4,a3 are all constants
q = 0.205
• Form of Q() is redefined as
• Q() = [ln(1 + e(qh)2)]1/2 [ln(e - (e - 2)e-(qh)2)]-1/2
PC EOS & MESA
• Low temperature high density region
• Carefully handles mixtures of carbon and
oxygen
• Accounts for all corrections to EOS laid out
earlier and more
• Ex => Inverse Beta Decay
• Can handle both classical and quantum Coulomb
crystals, Coulomb liquid interactions (weak or strong
coupling),…
• Default for  > 80